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Hughes B.D. — Random walks and random enviroments (Vol. 1. Random walks)
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Название: Random walks and random enviroments (Vol. 1. Random walks)Автор: Hughes B.D. Аннотация: Volume 1 can be read without reference to Volume 2. In Volume 1, I expect of the reader only a modest facility in classical analysis, including the theory of functions of a complex variable up to contour integration. Those elements of probability theory which are needed are introduced in Chapter 1 of Volume 1, so that although an acquaintance with elementary
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1995Количество страниц: 631Добавлена в каталог: 13.10.2005Операции: Положить на полку |
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'Good' solvent 416 Abel's means, summation by 216 Absolutely continuous function 34 Absorbing boundary, continuum diffusion 228 Absorbing boundary, one-dimensional random walk 113 Alexander — Orbach scaling law 317 336 Allowed step 139 Amplitude function, self-avoiding walks and polygons 485 Amplitude function, structure function of Weierstrass random walk 214 Analytic continuation of spin dimension in n-vector model 535 Analytic continuation, familiarity assumed 569 Analytic continuation, precluded 218 Analytical chaos 5 Appell's function of fourth kind 583 Appell's Theorem 582 Asymptotic expansion 589 Attractive fixed point 518 Autocorrelation time in Monte Carlo simulation of self-avoiding walk walk 465 Average value 36 Bachelier's chain equation 207 Bachelier's equation 209 Backbone of a self-avoiding walk 431 Backward equations 251 280 Bailey's theorem 583 Bernoulli numbers 573 Bernoulli's theorem 54 Berretti — Sokal algorithm 466 Bessel function of imaginary argument 579 Bessel Function, First Kind 577—578 Bessel function, integrals involving 580—581 Bessel function, modified 579 beta function 572 Bethe lattice, correlated walk on 179 Bethe lattice, effectively infinite-dimensional 23 Bethe lattice, introduced 22 Bethe lattice, Polya walk on 113 Bipartite lattice, random walk on 336 Body-centred cubic lattice, Green function evaluated 605 Body-centred cubic lattice, return probability 153 Body-centred cubic lattice, structure function for 140 Boltzmann distribution 229 390 Boltzmann's constant 390 Bond Green function 303 Bond of a lattice 19 Bond percolation threshold 307 Bond-to-site transformation 496 Box dimension 15 Bravais lattice, defined 22 Bravais lattice, random walk on 136 bridges 429 436 445 Brillouin zone, first 137 Brownian motion 1 8 14 48 189 Brownian motion, generating self-affine set 17 Bubble diagram 446 Cacti, defined 25 Cacti, random walk on 179 Cantor (ternary) set 7 Cardinality of random walk see "Number of distinct sites visited" Catalan's constant 508 Cauchy density in winding angle problem 79 Cauchy density, defined 37 Cauchy density, example of stable density 198 Cauchy density, Fourier transform of 42 Cayley tree see "Bethe lattice" Cellular automata 5 Central Limit Theorem, analogue for 359 Central Limit Theorem, Bernoulli's precursor of 54 Central Limit Theorem, Gram — Charlier refinements of 65 71 Central Limit Theorem, illustrated for one-dimensional Polya walk 64 Central Limit Theorem, introduced and stated 42—43 Central Limit Theorem, produces Gaussian density 189 Central Limit Theorem, sharp versions 197 Central Limit Theorem, three-dimensional version 91 Central Limit Theorem, two-dimensional version 70 Centre of mass of random walk 96 Chain equation 209 Chapman — Kolmogorov equation 207 209 Characteristic function 40 Chebyshev's inequality, stated and proved 44 Chebyshev's inequality, used 343 Clausen's Theorem, proved 582 Clausen's Theorem, stated 581 Clausen's Theorem, used 606 Coherent potential approximation 300 Collapse transition 515 Compact exploration 390 Compass dimension 15 Complementary events 31 Complete elliptic integral, an integral leading to 575 Complete elliptic integral, defined 575 Complete elliptic integral, evaluated via gamma functions 576—577 Complete elliptic integral, notational dangers 575 Complete elliptic integral, series expansion via Mellin transforms 592—593 Complete elliptic integral, series for 576 Complete elliptic integral, transformation of 576 Complete self-avoiding walks 507 Conditional mean first-passage time 126 Conditional probability 31 Conformal invariance 545 Connective constant for self-avoiding polygon 440 Connective constant for self-avoiding trail 495 Connective constant for self-avoiding walk 423 Connective constant, 'exact' for honeycomb lattice 426 428 436 481 544 Connective constant, estimates for dimension 490 Connective constant, estimates for three dimensions 483 Connective constant, estimates for two dimensions 482 Connective constant, inequalities for 426—436 Continuous-time random walk, analogue of Polya's Theorem 253 Continuous-time random walk, analysed by renormalization in one dimension 311 Continuous-time random walk, analysed by renormalization on Sierpinski lattice 314 Continuous-time random walk, analysed by renormalization on square lattice 313 Continuous-time random walk, diffusive 268 Continuous-time random walk, Montroll — Weiss model 243 Continuous-time random walk, Scher — Lax model 244 278—283 Continuous-time random walk, separable 244 Continuous-time random walk, subdiffusive 268 Continuous-time random walk, superdiffusive 268 Continuum double diffusion 225 Continuum resistance dimension 234 Convective diffusion equation 229 Convergence in probability 45 Convex function 45 Convex polygons (self-avoiding) 510 Convolution theorem for Fourier transform 40 Convolution theorem for Laplace transform 44 Convolution theorem for Mellin transform 591 Coordination number 20 Correction-to-scaling exponent 470 Correlated random walk 179 Correlation length for n-vector model, defined 528 Correlation length for n-vector model, exponents for 529 Countable 9 Covering lattice in Kasteleyn's walk problem 507 Covering lattice, defined 496 Covering lattice, significance in trail problem 496 Creeper, asymptotic behaviour 288 Creeper, defined 287 Creeper, model for turbulent dispersion 289 Critical amplitudes for self-avoiding walk 486 Critical exponent, n-vector model 525 527 529 Critical exponent, number of self-avoiding polygons 444 Critical exponent, number of self-avoiding trails 496 Critical exponent, number of self-avoiding walks 426 Critical exponent, structure function of Weierstrass random walk 213 Critical exponent, superdiffusive motion 199 Critical point in magnetic model 221 Critical point in method of steepest descent 100 Critical temperature in mean field treatment of magnetic model 533 Critical temperature in n-vector model 524 Cumulant generating function 374 Cumulants 373 Cumulative distribution function 33 Darboux's Theorem, stated 118 Darboux's Theorem, used 337 339 341 475 Debye's method 98 Decimated lattice 309 Decimated site 309 Decimation in renormalization analysis of random walk 309 Defective site in random walk 165 Delta function, Dirac's 9 28 33 Delta function, one-sided 33 Density of states, integrated 28 422 Density of states, vibrational 422 Denumerable 9 Detailed balance condition, generalized master equation 292 Detailed balance condition, master equation 290 Detailed balance condition, self-avoiding walk simulation 467 Deterministic chaos 5 Deterministic fractal 14 Devil's staircase 34 Dhar's formula 335 Diagmma function 572 Diamond lattice, Green function for 609 Diamond lattice, return probability 153 Dielectric constant, frequency-dependent complex 259 Dielectric constant, static 259 Diffusion constant for one-dimensional continuous-time random walk 268 Diffusion constant in diffusion equation 191 Diffusion equation with constant drift 191 Diffusion equation, derived in Fourier transform space 194—196 Diffusion equation, derived in real space 190—191 Diffusion equation, first-passage times 193 Diffusion equation, solved by Fourier transforms 191—193 Diffusion resistance 233 Diffusion-limited aggregation 398 Diffusive behaviour, loosely defined 284 Diffusive random walk 268 Dimension of a lattice 20 Dimension, based on resistance 233 Dimension, box 15 Dimension, compass 15 Dimension, continuum harmonic 234 Dimension, continuum resistance 234 Dimension, divider 15 Dimension, effective for random walk 157 Dimension, first-passage time 231 319 Dimension, fractal 13 210 336 Dimension, fractal, of a lattice 27 Dimension, fractal, of a self-avoiding walk 420—421 Dimension, fracton 30 317 336 Dimension, gap 17 Dimension, harmonic 29 30 335 Dimension, Hausdorff — Besicovitch 8 210 421 Dimension, intuitive 6 Dimension, local fractal 421 Dimension, mass 15 421 Dimension, reservations about terminology 9 Dimension, scaling 10 Dimension, scaling of a lattice 27 Dimension, spectral 30 336 Dimension, topological 7 Dimension, upper critical 422 534 541 Dimensional multiplicity 16 Dimensionally discordant set 13 Dimer problem 507 Dirac delta function 9 28 33 Directed lattices 20 Directed self-avoiding walks 491 Discrete Fourier Transform 40 Discrete Tauberian Theorem, analogue for Fourier series 160 Discrete Tauberian Theorem, misleading idea of accuracy 338 Discrete Tauberian Theorem, stated 118 Discrete Tauberian Theorem, used 128 130 141 334—336 338 340 355 362 368 474 490 539 601 603 Dispersion, relation 28 Dispersion, relative 271 Distribution function 33 Divider dimension 15 Domb — Joyce model 395 Double diffusion 225 Drag coefficient 229 Drunkard 1 Duplication formula 570 Dvoretzky — Erdos Lemma 332 Dyadic see "Tensor" Dyadic, use of boldface symbol 117 Effective dimension of a random walk 157 Effective medium approximation, basic formalism 300—304 Effective medium approximation, history 300 Effective medium approximation, single bond 304—308 Eigenfunction in lattice vibrations 134 Eigenfunction of operator 230 386 Eigenvalue in renormalization 221 Eigenvalue of operator 231 Eigenvalue of tensor A 143 355 Einstein — Smoluchowski relation for continuum diffusion 230 Enumeration techniques for self-avoiding walk see "Series expansions" Equilibrium configurations 391 Ergodicity of Monte Carlo algorithms 465 Euclidean space 6 Euler transformation 476 Euler walks 507 Euler's constant 570 573 Events, complementary 31 Events, independent 31 Events, introduced 30 Events, mutually exclusive 31 Excluded volume problem, defect of random flight model 98 Excluded volume problem, self-avoiding walk model 414 Expectation 36 Expected value 36 Face-centred cubic lattice, Green function evaluated 607 Face-centred cubic lattice, return probability 153 Face-centred cubic lattice, structure function for 140 Fick's law 228 Finitely ramified 514 First Brillouin zone 28 137 First-passage time density for continuous-time random walk 252 First-passage time density for diffusion equation 193 First-passage time dimension for continuum diffusion 231 First-passage time for diffusion equation 193 First-passage time for lattice walk 120 Fisher — Flory formula 451 Fisher's Lemma 455 Fixed point of an iterated mapping 518 Flory, P.J. 415 Fluctuation-dissipation theorem, 'proved' 528 Fluctuation-dissipation theorem, used 530 Forbidden site 168 Forward equations 251 280 Fourier sine transform 87 93 Fourier transform and moments 41 Fourier transform, defined 39 Fourier transform, discrete 40 Fourier — Bessel series, applied to Pearson's walk 72 Fourier — Bessel series, properties 578 Fractal dimension 336 Fractal dimension of Brownian motion trail 48 210 Fractal dimension of random walk trail 210 Fractal dimension of Weierstrass random walk trail 212 Fractal dimension, synonymous with Hausdorff — Besicovitch or scaling 13 Fractal lattice 25 Fractal sets 13 Fractal time 246—247 Fractals 13 Fracton dimension 30 317 336 Free energy of n-vector model 523 Free group 23 Gamma density, infinitely divisible but not stable 209 Gamma density, sum of random variables with exponential density 246 Gamma function, defined 569
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